The Dirichlet Multinomial Distribution as a Magazine Exposure Model

Leckenby, John D.; Kishi, Shizue

doi:10.1177/002224378402100110

Cited by 32 publications

(15 citation statements)

References 12 publications

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“…The population's reading heterogeneity is modeled rather simply, whereas a more complicated model could give an even better fit. Lastly, an examination of the Markov models' stationarity assumption seems necessary, perhaps along similar lines to that conducted on the BBD by Sabavala and Morrison (1981 RER and EPOR were also used by Leckenby andKishi (1982a, 1984).…”

Section: Appendix: Initial Parameter Estimatesmentioning

confidence: 99%

“…The Markov mixture model can also be used as a marginal distribution when estimating multidimensional ED's (Danaher 1988a). In addition, this model can be applied to television viewership and newspapers (whose readers exhibit high loyalty) and to consumer purchasing behavior ule whose observed sample ED is already known (Chandon 1976;Danaher 1988a,b;Leckenby andKishi 1982a, 1984;Rust and Leone 1984). We use Danaher's (1988a) goodness-of-fit criteria based on reach, GRP's, and the absolute difference of fitted and observed distributions.…”

Section: Appendix: Initial Parameter Estimatesmentioning

confidence: 99%

“…In fact the BBD is the most popular nonproprietary magazine model in use (Leckenby and Kishi 1982b). Now express S as the sum of k indicator random variables; Xi = 1 if the ith issue of the magazine is read and°o therwise.…”

mentioning

confidence: 99%

“…Let S be the number of exposures, and let P* be the probability that an individual is exposed to anyone ad. The distribution of S is usually called the exposure distribution (ED) (Danaher 1988a,b;Leckenby andKishi 1982a, 1984). Ehrenberg (1975) and Danaher (1988b) assumed that S IP* -bin(k, P*).…”

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confidence: 99%

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A Markov Mixture Model for Magazine Exposure

Danaher

1989

Journal of the American Statistical Association

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A magazine-exposure model that mixes Klotz's (1973) dependent Bernoulli-trials model for nonsubscribers with a degenerate distribution for subscribers is proposed. Let X, = 1 if a person reads an issue of a particular magazine and 0 otherwise. Klotz's parameterization is Pr(X i = 1) = P. and Pr(X, = 1 I x'-I = 1) = Afor i = 1, ... , k. Using the Markov assumption he obtains the joint distribution of R =~f=2 Xi-IX" S =~f=l X" and T = XI + Xb of which we are interested in the marginal distribution of S, the total number of issues a person reads. It is expected that p will be low for nonsubscribers but high for subscribers, so this heterogeneity is modeled by mixing Klotz's Markov model with a point mass of magnitude It at the point S = k. Maximum likelihood estimates of p , A, and It are used to fit the Markov mixture model to 40 magazines from a large print-media survey. The proposed model is shown to give a much better fit to these data than the beta-binomial model, the most popular nonproprietary magazine model, and a generalization of the beta-binomial model. where°< OJ < 1 and hIis the indicator function. Danaher (1988b) showed that the MBBD usually gives a better fit to magazine ED's than the BBD, at the expense of an additional parameter.The BBD and MBBD work reasonably well in practice for magazine ED's (Chandon 1976;Danaher 1988b;Greene and Stock 1967;Metheringham 1964), although the BBD has been shown to overestimate odd-numbered exposures by Schreiber (1969) and Chandon (1976). In fact the BBD is the most popular nonproprietary magazine model in use (Leckenby and Kishi 1982b). Now express S as the sum of k indicator random variables; Xi = 1 if the ith issue of the magazine is read and°o therwise. Klotz (1973) defined Assuming the Markov property, Klotz developed a Markov model for Bernoulli trials with dependence. He derived the exact joint distribution of (R, S, T), where R =~f=2 Xi-lXi, S =~f=l Xi' and T = Xl + Xi; and thenused this distribution to model monthly rainfall in Madison, Wisconsin. The distribution of just S was derived by Gabriel (1959) using a different parameterization. Klotz (1972) also used (R, S, T) to model the dependency of sex on that of the previous child. Chandon (1976) mentioned but did not fully develop a Markov model for magazine ED's using Klotz's (1973) model. For magazine readership p is a measure of popularity and A is a measure of the dependency between successive trails. Sabavala andMorrison (1977, 1981) call Athe "loyalty rate." The parameters p and A are aggregate population parameters; that is, each person has the same p and A. However, the population is expected to be heterogeneous with respect to reading behavior, being divided into two groups. The first group is called nonsubscribers, who rarely or never read a particular magazine, and the second is called subscribers, who always read the magazine. The population tends to cluster into these two groups, in which case p < A (Klotz 1973). In Section 2 I allow for this heterogeneity by mixing a Markov model for nonsub...

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Section: Appendix: Initial Parameter Estimatesmentioning

confidence: 99%

Section: Appendix: Initial Parameter Estimatesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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A Markov Mixture Model for Magazine Exposure

Danaher

1989

Journal of the American Statistical Association

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“…models have been developed (cf. Chandon 1976;Danaher 1988E;Leckenby and Kishi 1984;Rust and Leone 1984), with varying degrees of accuracy.…”

Section: Danahermentioning

confidence: 99%

Simulating Media Exposure Distributions

Danaher

1989

Communications in Statistics - Simulation and Computation

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Estimation for dirichlet mixed models

Leeds

Gelfand

1989

Naval Research Logistics

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Dirichlet mixed models find wide application. Estimation is usually achieved through the method of moments. Here we present an iterative hybrid algorithm for obtaining the maximum likelihood estimate employing both modified Newton‐Raphson and E‐M methods. This successful MLE algorithm enables calculation of a jack‐knife MLE. Simulation comparison of the three estimates is provided. The MLE substantially improves upon the moments estimator particularly with increasing dimension. The jack‐knife MLE in turn offers dramatic improvement over the MLE.

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The Dirichlet Multinomial Distribution as a Magazine Exposure Model

Cited by 32 publications

References 12 publications

A Markov Mixture Model for Magazine Exposure

A Markov Mixture Model for Magazine Exposure

Simulating Media Exposure Distributions

Estimation for dirichlet mixed models

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